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Loewner’s theorem for maps on operator domains

Part of: Measures, integration, derivative, holomorphy Special classes of linear operators

Published online by Cambridge University Press:  16 May 2022

Michiya Mori  and
Peter Šemrl
Michiya Mori
Affiliation:
Interdisciplinary Theoretical and Mathematical Sciences Program (ITHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan e-mail: michiya.mori@riken.jp
Peter Šemrl*
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia and Department of Mathematics, Institute of Mathematics, Physics, and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia
*
e-mail: peter.semrl@fmf.uni-lj.si
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Abstract

The classical Loewner’s theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.

Keywords

Loewner’s theoremoperator domainlocal order isomorphismorder embeddingbiholomorphic mapgeneralized upper half-plane

MSC classification

Primary: 46G20: Infinite-dimensional holomorphy
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B49: Transformers, preservers (operators on spaces of operators)
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 3 , June 2023 , pp. 912 - 944
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The second author was supported by grants N1-0061, J1-8133, and P1-0288 from the ARRS, Slovenia.

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